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<p>We develop a nonparametric, semimartingale optimal transport, calibration methodology for local volatility models with stochastic interest rates. The method finds a fully calibrated model which is closest, in a way defined by a general cost function, to a given reference model. We establish a general duality result which allows to solve the problem by optimising over solutions to a second-order fully nonlinear Hamilton–Jacobi–Bellman equation. Our methodology is analogous to Guo et al. (SIAM J. Financ. Math. 13:1–31, <CitationRef CitationID="CR31">2022</CitationRef>; Math. Finance 32:46–77, <CitationRef CitationID="CR34">2022</CitationRef>), but features a novel element of solving for discounted densities, or sub-probability measures. As an example, we apply the method to a sequential calibration problem, where a Vašíček model is already given for the interest rates and we seek to calibrate a stock price’s local volatility model with a volatility coefficient depending on time, the underlying and the short rate process, and the two processes driven by possibly correlated Brownian motions. The equity model is calibrated to any number of European option prices.</p>

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Calibration of local volatility models with stochastic interest rates using optimal transport

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月23日

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  • Benjamin Joseph,
  • Grégoire Loeper,
  • Jan Obłój

摘要

We develop a nonparametric, semimartingale optimal transport, calibration methodology for local volatility models with stochastic interest rates. The method finds a fully calibrated model which is closest, in a way defined by a general cost function, to a given reference model. We establish a general duality result which allows to solve the problem by optimising over solutions to a second-order fully nonlinear Hamilton–Jacobi–Bellman equation. Our methodology is analogous to Guo et al. (SIAM J. Financ. Math. 13:1–31, 2022; Math. Finance 32:46–77, 2022), but features a novel element of solving for discounted densities, or sub-probability measures. As an example, we apply the method to a sequential calibration problem, where a Vašíček model is already given for the interest rates and we seek to calibrate a stock price’s local volatility model with a volatility coefficient depending on time, the underlying and the short rate process, and the two processes driven by possibly correlated Brownian motions. The equity model is calibrated to any number of European option prices.